3 The structures of simple solids

  Answer  We need to identify the displacements that, when applied to the entire cell, result in every atom 

arriving at an equivalent location (same atom type with the same coordination environment). In this case, 

the displacements (0, 

+½, +½), (+½, +½, 0), and (+½, 0, +½), where +½ in the  x ,   y ,  or   z   coordinate 

represents a translation along the appropriate cell direction by a distance of  a 

/2,  b /2, or  c /2 respectively, 

have this effect. For example, starting at the labelled Zn 

2

+ 

  ion towards the near bottom left-hand corner of 

the unit cell (the origin), which is surrounded by four S 

2

− 

  ions at the corners of a tetrahedron, and applying 

the translation (

+½, 0, +½), we arrive at the Zn 

2

+ 

  ion towards the top front right-hand corner of the unit 

cell, which has the same tetrahedral coordination to sulfur. Identical translational symmetry exists for all 

the ions in the structure. These translations correspond to those of the face-centred lattice, so the lattice 

type is F. 

  Self-test  3.1  Determine the lattice type of CsCl ( Fig.  3.7  ).   

     (b)   Fractional  atomic  coordinates  and  projections   

  Key  point:  

Structures may be drawn in projection, with atom positions denoted by fractional 

coordinates. 

 The position of an atom in a unit cell is normally described in terms of   fractional coordi-

nates  , coordinates expressed as a fraction of the length of a side of the unit cell. Thus, the 

position of an atom, relative to an origin (0,0,0), located at  xa  parallel to  a ,  yb  parallel to 

 b , and  zc  parallel to  c  is denoted ( x , y , z ), with 0 ≤  x ,  y ,  z  ≤ 1. Three-dimensional representa-

tions of complex structures are often diffi cult to draw and to interpret in two dimensions.   

A clearer method of representing three-dimensional structures on a two-dimensional sur-

face is to draw the structure in projection by viewing the unit cell down one direction, 

typically one of the axes of the unit cell. The positions of the atoms relative to the projec-

tion plane are denoted by the fractional coordinate above the base plane and written next 

to the symbol defi ning the atom in the projection. If two atoms lie above each other, then 

both fractional coordinates are noted in parentheses. For example, the structure of body-

centred tungsten, shown in three dimensions in  Fig.  3.8  a, is represented in projection in 

 Fig.  3.8  b.   

   E X A MPLE  3. 2 

Drawing a three-dimensional representation in projection   

  Convert the face-centred cubic lattice shown in  Fig.  3.5   into a projection diagram. 

  Answer   We need to identify the locations of the lattice points by viewing the cell from a position 

perpendicular to one of its faces. The faces of the cubic unit cell are square, so the projection diagram 

viewed from directly above the unit cell is a square. There is a lattice point at each corner of the unit cell, so 

the points at the corners of the square projection are labelled (0,1). There is a lattice point on each vertical 

face, which projects to points at fractional coordinate ½ on each edge of the projection square. There are 

lattice points on the lower and on the upper horizontal face of the unit cell, which project to two points 

at the centre of the square at 0 and 1, respectively, so we place a fi nal point in the centre of a square and 

label it (0,1). The resulting projection is shown in  Fig.  3.9  .